Find the domain and range of a linear, quadratic, cubic, square root, log, reciprocal or exponential function by picking its form and coefficients.
Domain and range
The domain is the set of valid real inputs. The range is the set of outputs the function actually attains. Optional input bounds are intersected with the formula’s real domain before the output range is found.
| Function | Domain | Range |
|---|---|---|
| ax + b; a(x − h)³ + k | All real x | All real y |
| a(x − h)² + k; a|x − h| + k | All real x | [k, ∞) if a > 0; (−∞, k] if a < 0 |
| a√(x − h) + k | [h, ∞) | [k, ∞) if a > 0; (−∞, k] if a < 0 |
| a ln(x − h) + k | (h, ∞) | All real y |
| a/(x − h) + k | All real x except h | All real y except k |
| a·b^(x − h) + k, b > 0, b ≠1 | All real x | (k, ∞) if a > 0; (−∞, k) if a < 0 |
Example
For f(x) = 2(x − 3)² − 4 on 1 ≤ x ≤ 5, the vertex x = 3 is included. Its minimum is −4, and both endpoints give 4. The domain is [1, 5] and the range is [−4, 4].
Frequently asked questions
What do parentheses and brackets mean?
[ ] includes a finite endpoint; ( ) excludes it. Infinity is always excluded. A single attained value is written {value}.
What happens if a = 0?
The range becomes a singleton if any valid input remains. Formula restrictions still apply: ln(x − h) needs x > h and a reciprocal still excludes x = h. Exponential base 1 also produces a constant.
Is the range estimated from sampled graph points?
No. The calculator uses endpoint limits and monotonic behavior, plus a quadratic vertex or absolute-value corner when included. Supported finite values use numerical arithmetic; results outside its representable range produce an error.
References
